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Mathematics 602 A. Hulpke Homework (due Feb 9) 1) (Keeler’s theorem) Show that any permutation π ∈ S n can be written as a product of distinct transpositions (i.e. permutations (a, b)) from S n+2 , none of which lies in S n . 2) Classify the subgroups of D 8 × D 8 that project surjectively on both copies of D 8 . 3) Let G = GLn (q) = {M ∈ Fn×n ∣ det M =/ 0}. q a) Show that there is a faithful homomorphism G → S q n . b) Show that (action on 1-dimensional subspaces) there is a homomorphism πG → S q n −1 and that ker π = Z(G) = q−1 F∗q ⋅ 1n×n . c) Determine the number of bases of Fnq and show that this is the order of G. d) Show that if q = p a (we call p the defining characteristic) the upper triangular matrices with diagonal 1 form a p-Sylow subgroup of G. e) Now let H = SLn (q) = {M ∈ G ∣ det M = 1}. Determine the orders of H, ker π ∩ H and ⟨ker π, H⟩. e) In which cases is G = Z(G) × H? 4) Let p be a prime and G ≤ S p . a) Show that if G is transitive and ⟨1⟩ =/ N ⊲ G, then N must be transitive. b) Let G ≤ S p such that G is generated by two p-cycles. Show that G must be simple. (Hint: Use a) to show that any nontrivial normal subgroup of G is transitive. Then use the Sylow theorems to show that it must contain the generators of G. c) For p = 5, p = 7, and p = 11 determine the simple groups obtained this way. 5) Let G ≤ S n transitive, and Ω = {1, . . . , n}. Show that there is a bijection between the orbits of G on Ω × Ω (acting as (ω, δ) g = (ω g , δ g )), and the orbits of StabG (1) on Ω. We say that G is doubly transitive if there is just one orbit (besides {(ω, ω) ∣ ω ∈ Ω}). 6) Let n be a positive integer, p a prime and k = log p (n). Consider n written as a sum of p-powers k n = ∑ ai pi i=0 Show that the power of p that divides n! is exactly p ⎛k ⎛ i−1 j ⎞⎞ ∑ ai ⋅ ∑ p ⎝ i=1 ⎝ j=0 ⎠⎠ (i.e. this is the order of a p-Sylow subgroup of S n ). 7) The commutator of x, y ∈ G is defined as [x, y] = x −1 y−1 x y. For groups A, B ≤ G we define [A, B] = ⟨[a, b] ∣ a ∈ A, b ∈ B⟩. (That is ever element of [A, B] is a product of commutators, but not neccessarily a commutator itself.) a) Show that if A, B ⊲ G, then [A, B] ≤ A ∩ B and [A, B] ⊲ G. b) Show that if A ⊲ G then G/A is abelian if and only if G ′ = [G, G] ≤ A. c) A group is called perfect if G = [G, G] = G ′ . (Clearly every simple nonabelian group is perfect.) Show that SL2 (5) is perfect, but not simple.